This was a lot of fun to play around with! Thanks!

'I'd better turn my attention to trimming the hedges and other things my wife thinks are important. Fractals don't impress her much.'

When I show my Mathematica-generated Maths Art to most friends I suppose they just think "oh just that's another one of those thingies Darren seems to like making all of the time, he seems to find every single one of them particularly special and interesting, but they all look the same to me". But they aren't. Each one is special.

Okay, okay. I'll stop after this. But each new image is more exciting than the last. Like the image above, this one has two layers put together with ImageCompose. The bottom layer is made by the resource function MandelbrotSetRemap with a custom mapping function based on standard deviation. The variety of custom mappings seems boundless when you look at all the Wolfram functions for lists and numbers. The top layer uses the default mapping in MandelbrotSetRemap, but the color function has transparency to let the bottom layer through.

Here's the code. Be patient, custom mapping functions take a lot of time to render and this code renders at a very large scale and then shrinks the image to avoid the jaggies:

msr = ResourceFunction["MandelbrotSetRemap"];
myMapping1 = 
  Function[{c, center, corner, maxIterations}, Module[{list},
    list = NestWhileList[#^2 + c &, 0, Abs[#] <= 4 &, 1, maxIterations];
    Total[Arg[list]] * StandardDeviation[list]]
  ];
img1 = msr[-.875 + .256 I, 78, MaxIterations -> 32, "MappingFunction" -> myMapping1, 
  ColorFunction -> ColorData[{"PlumColors", "Reversed"}], ImageSize -> 2400];
colFn = (Blend[{{0, Transparent}, {.25, Transparent}, {.45, Black}, {.97, LightPurple}, {1, Darker[Blue, .8]}}, #1] &);
img2 = msr[-.875 + .256 I, 78, MaxIterations -> 72, ColorFunction -> colFn, ImageSize -> 2400];
ImageResize[ImageCompose[img1, img2], 600]

Yes! Don't stop Mark. Would be great if you could also include the code you used to generate the images so others can use it as a starting point for their own explorations.

And here's another image. This one combines two layers rendered from the resource function MandelbrotSetRemap with no additional filtering.

Here's the code:

msr = ResourceFunction["MandelbrotSetRemap"];
colFn1 = (Blend[{{0, Transparent}, {.35, Black}, {.4, Transparent}, {1,  Black}}, #1] &);
layer1 = msr[-.8298 - .2281 I, 1103, MaxIterations -> 96, ColorFunction -> colFn1, ImageSize -> {1200, 800}];
colFn2 = (Blend[{{0, Red}, {.2, Darker[Red]}, {.6, Orange}, {1, Yellow}}, #1] &);
layer2 = msr[-.8298 - .2281 I, 1103, MaxIterations -> 44, MappingFunction -> "LineOrbitTrap", ColorFunction -> colFn2, ImageSize -> {1200, 800}];
ImageCompose[layer2, layer1]

This is quite beautiful and I hope some form makes its way to the Wolfram Function Repository.

And ignore those who object to the artistry, they're just being Mandelbrats.

The speed is not too bad. It takes 10 or so seconds to create the first couple of custom images on my machine. You could probably get a good speed boost using Compile though.

Here is an image I made with the new resource function MandelbrotSetRemap. It has been further stylized with other Wolfram Language filters.

Here's the code for this image:

msr = ResourceFunction["MandelbrotSetRemap"];
img1 = msr[.333, 32, MaxIterations -> 128, MappingFunction -> "ModTrailings", ImageSize -> {1200, 800}];
img2 = msr[.333, 32, MaxIterations -> 64, MappingFunction -> "Decomposition", 
  ColorFunction -> ColorData["BrightBands"], ImageSize -> {1200, 800}];
ImageAdjust[ImageEffect[ImageMultiply[{img1, img2}], {"EdgeStylization"}]]